Consistent model selection based on parameter estimates

We consider model selection based on estimators that are asymptotically normal. Such a method can be applied to the context of estimating equations, since a complete specification of the probability model or likelihood function is not required. We construct a cost function for the models in consideration, and show that the minimizer of the cost function is a consistent estimator of the model. Despite the absence of a likelihood function, the cost function is shown to be related to an approximate posterior probability conditional on the parameter estimates, which enables a Bayesian-type evaluation of all candidate models and not just to present one best choice. The proposed method is modular and easily adapted to different problems, since only one set of estimates of the parameters and asymptotic variance is needed as the input, which can be obtained from very different estimation techniques for very different models, both linear and nonlinear. We also show that by ranking Z-statistics, the scope of model searching can be reduced to achieve computing efficiency. We provide data analysis examples from two clinical trials and illustrate these variable selection techniques in the contexts of partial likelihood analysis and generalized estimating equations. A third example of used automobile prices illustrates an application of the methodology in selecting graphical models.